Finite Group Databasebeta

← #27446677

Subgroups — #27446677

Order 13568.

838 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810930C211yesmaximal, characteristic, complemented
2C22×#13133150C211yesmaximal, characteristic
3#20278439C211yesmaximal, characteristic
4#20276263C211yesmaximal, characteristic, complemented
5#20276206C211yesmaximal, characteristic, complemented
6#20276110C211yesmaximal, characteristic, complemented
7#20276053C211yesmaximal, characteristic, complemented
8#20275483C211yesmaximal, characteristic, complemented
9#20275286C211yesmaximal, characteristic, complemented
10#20275139C211yesmaximal, characteristic
11#20275023C211yesmaximal, characteristic
12#20274969C211yesmaximal, characteristic, complemented
13#20274908C211yesmaximal, characteristic
14#20274895C211yesmaximal, characteristic
15C53×#2360C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19#16810930—24no—
20C2×#13133150—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133141C2211yescharacteristic
32C2×#13133141C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34#16810565—21no—
35#16810557—21no—
36#16810521C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810495—21no—
39#16810489C2211yescharacteristic
40#16810483C2211yescharacteristic
41#16810481—21no—
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#389—21no—
55C53×#389C2211yescharacteristic
56C53×#387—21no—
57C53×#385—21no—
58C4×C4⋊C4×C53C2211yescharacteristic
59C4×C22⋊C4×C53—21no—
60C4×C22⋊C4×C53C2211yescharacteristic
61C23×Dic53—21no—
62C22×#88322—22no—
63C22×#88322—22no—
64C22×#88322—22no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133209—22no—
70#13133209—41no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53—41no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83#13133150—22no—
84#13133150—22no—
85#13133150—22no—
86#13133150D414yes—
87#13133148—22no—
88#13133148—41no—
89C4⋊#88322—22no—
90C4⋊#88322—22no—
91#13133141—22no—
92#13133141—22no—
93#13133141—22no—
94#13133141—41no—
95C22⋊#88322—22no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101C22×Q8×C53C2311yescharacteristic
102C23×C4×C53—21no—
103C23×C4×C53C2311yescharacteristic
104C22⋊Q8×C53—22no—
105C22⋊Q8×C53Q814yes—
106C4×Q8×C53—22no—
107C4×Q8×C53—41no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—22no—
113C2×C4⋊C4×C53—41no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53D411yescharacteristic
119C2×C22⋊C4×C53D411yescharacteristic
120C2×C42×C53—21no—
121C2×C42×C53—21no—
122C2×C42×C53C2311yescharacteristic
123C2×C42×C53C2311yescharacteristic
124#95×C53—21no—
125#95×C53—21no—
126#95×C53—21no—
127#95×C53—21no—
128#95×C53D411yescharacteristic
129#95×C53D411yescharacteristic
130C24×C53C2×D411yescharacteristic
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53C2×Q812yes—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—42no—
146C22×C4×C53—42no—
147C22×C4×C53—81no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×Q812yes—
154C22×C4×C53#4112yes—
155C22×C4×C53C2411yescharacteristic
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—42no—
161C22×Dic53—42no—
162C22×Dic53—81no—
163C4⋊C4×C53—24no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53—81no—
171C4⋊C4×C53C2×D412yes—
172C4⋊C4×C53C2×Q812yes—
173C22⋊C4×C53—24no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—42no—
181C42×C53—81no—
182C22⋊Dic53—42no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—42no—
191C4⋊Dic53C2×D412yes—
192#11719919—42no—
193#11719919—42no—
194#11719919—81no—
195#11719919—81no—
196C2×#88322—28no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322C2×D412yes—
208C2×#88322#4112yes—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53C22⋊Q812yes—
214C23×C53#13512yes—
215C23×C53#13712yes—
216C23×C53C22×D411yescharacteristic
217Q8×C53—22no—
218Q8×C53—22no—
219Q8×C53—41no—
220Q8×C53—41no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—84no—
252C2×C4×C53C22⋊Q812yes—
253C2×C4×C53C22×D411yescharacteristic
254C2×C4×C53C22×Q811yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53H(5,𝔽2)11yescharacteristic
258C2×C4×C532−1+411yescharacteristic
259C2×Dic53—22no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—84no—
272#88322—22no—
273#88322—22no—
274#88322—41no—
275#88322—41no—
276#26344—531nomaximal, Sylow
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53#47914yescomplemented
284C22×C53C2×C22⋊Q811yescharacteristic
285C22×C53D4211yescharacteristic
286C22×C53#54711yescharacteristic
287C22×C53C2×#13511yescharacteristic
288C22×C53C2×#13711yescharacteristic
289C22×C53#57611yescharacteristic
290C22×C53#57811yescharacteristic
291C4×C53—22no—
292C4×C53—41no—
293C4×C53—42no—
294C4×C53—41no—
295C4×C53—42no—
296C4×C53—41no—
297C4×C53—41no—
298C4×C53—42no—
299C4×C53—81no—
300C4×C53#52511yescharacteristic
301C4×C53#53911yescharacteristic
302C4×C53D4×Q811yescharacteristic
303C4×C53#55411yescharacteristic
304Dic53—42no—
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308C2×#479—531no—
309C22×#107—531no—
310#2442—531no—
311#2360D5311yescharacteristic
312#2097—531no—
313#2086—531no—
314#2068—531no—
315#2059—531no—
316#1949—531no—
317#1918—531no—
318#1892—531no—
319#1872—531no—
320#1863—531no—
321#1853—531no—
322#1850—531no—
323C2×C53—24no—
324C2×C53#244911yescharacteristic
325C2×C53C2×#47911yescharacteristic
326C2×C53#308311yescharacteristic
327C2×C53#327011yescharacteristic
328C2×C53#328011yescharacteristic
329C2×C53#333611yescharacteristic
330C2×C53#334611yescharacteristic
331C23×C8—531no—
332C2×#125—531no—
333C2×C22⋊Q8C2×D5311yescharacteristic
334C2×C4×Q8C2×D5311yescharacteristic
335C22×C4⋊C4—531no—
336C22×C4⋊C4C2×D5311yescharacteristic
337#479—1064no—
338C2×#107—531no—
339C2×#107—531no—
340C2×#107—531no—
341C2×#107—531no—
342C2×#107—1064no—
343C2×#106—531no—
344C2×#106—531no—
345C2×C4⋊C8—531no—
346C2×C4⋊C8—531no—
347C2×#103—531no—
348C2×#103—531no—
349C2×#103—531no—
350C2×#103—531no—
351C2×C22⋊C8—531no—
352#400—1061no—
353#398—1061no—
354#391C2×D5311yescharacteristic
355#389—21no—
356#389—531no—
357#389—531no—
358#389C2×D5311yescharacteristic
359#387—21no—
360#385—21no—
361#385—1061no—
362#382—531no—
363#380—531no—
364#380—1061no—
365C4×C4⋊C4C2×D5311yescharacteristic
366C4×C22⋊C4—21no—
367C4×C22⋊C4C2×D5311yescharacteristic
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#336—531no—
375#336—531no—
376C53#2634411yescharacteristic, complemented, Sylow
377C22×Q8C22×D5311yescharacteristic
378C23×C4—21no—
379C23×C4—1061no—
380C23×C4C22×D5311yescharacteristic
381C22×C8—531no—
382C22×C8—531no—
383C22×C8—531no—
384C22×C8—531no—
385C22×C8—1062no—
386C22×C8—1062no—
387C22×C8—1062no—
388#125—1062no—
389#125—2121no—
390C22⋊Q8—22no—
391C22⋊Q8Dic10614yes—
392C4×Q8—22no—
393C4×Q8—41no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—41no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—1062no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—2121no—
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4C22⋊D5311yescharacteristic
417C2×C22⋊C4C22⋊D5311yescharacteristic
418C2×C42—21no—
419C2×C42—21no—
420C2×C42C22×D5311yescharacteristic
421C2×C42C22×D5311yescharacteristic
422#107—534no—
423#107—1062no—
424#107—1062no—
425#107—1062no—
426#106—1062no—
427#106—2121no—
428C4⋊C8—1062no—
429C4⋊C8—1062no—
430#103—1062no—
431#103—1062no—
432#103—1062no—
433#103—2121no—
434C22⋊C8—1062no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95C22⋊D5311yescharacteristic
445#95C22⋊D5311yescharacteristic
446C24C2×C22⋊D5311yescharacteristic
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8C2×Dic10612yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—81no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—4241no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4C2×Dic10612yes—
472C22×C4#1171994512yes—
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C23×D5311yescharacteristic
479C2×C8—532no—
480C2×C8—532no—
481C2×C8—1068no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—81no—
500C4⋊C4—532no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—4241no—
509C4⋊C4—4241no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4C2×Dic10612yes—
514C4⋊C4D4×D5312yes—
515C22⋊C4—24no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—2122no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—81no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23#1313319612yes—
530C23#1313323712yes—
531C23#1313324812yes—
532C23C22×C22⋊D5311yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—41no—
536Q8—41no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—84no—
568C2×C4—1062no—
569C2×C4—4244no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4#1313319612yes—
582C2×C4C2×D4×D5311yescharacteristic
583C2×C4C22×Dic10611yescharacteristic
584C2×C4#41×D5311yescharacteristic
585C2×C4C2×#1171994511yescharacteristic
586C2×C4#1313326211yescharacteristic
587C2×C4#1313326411yescharacteristic
588C8—1062no—
589C8—1062no—
590C8—2121no—
591C8—2121no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22#1681093014yescomplemented
599C22C2×#1313319611yescharacteristic
600C22D4×C22⋊D5311yescharacteristic
601C22#1681127211yescharacteristic
602C22C2×#1313323711yescharacteristic
603C22C2×#1313324811yescharacteristic
604C22#1681142611yescharacteristic
605C22#1681143811yescharacteristic
606C4—22no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—42no—
612C4—41no—
613C4—42no—
614C4—81no—
615C4—4241no—
616C4—4241no—
617C4—4241no—
618C4—2122no—
619C4#1681113211yescharacteristic
620C4#1681121511yescharacteristic
621C4D4×Dic10611yescharacteristic
622C4#1681133211yescharacteristic
623C2—24no—
624C2#2027851811yescharacteristic
625C2C2×#1681093011yescharacteristic
626C2#2028456611yescharacteristic
627C2#2028628611yescharacteristic
628C2#2028639611yescharacteristic
629C2#2028707711yescharacteristic
630C2#2028719411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2360C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2360D53
hypocenterC53#26344
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16810930C2semidirect
4. #20276263C2semidirect
5. #20276206C2semidirect
6. #20276110C2semidirect
7. #20276053C2semidirect
8. #20275483C2semidirect
9. #20275286C2semidirect
12. #20274969C2semidirect
283. C22×C53#479semidirect
376. C53#26344semidirect
598. C22#16810930semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168109302
C22×#131331502
#202784392
#202762632
#202762062
#202761102
#202760532
#202754832
#202752862
#202751392
#202750232
#202749692
#202749082
#202748952
C53×#23602
#2634453